$\subset$ vs $\subseteq$ when *not* referring to strict inclusion Inspired by the confusion in the comments on this question:
I always thought that the standard was to read $\subset$ as "is a strict subset of", and $\subseteq$ could mean proper or improper inclusion.
Was I wrong?
 A: This is a very troubling issue with notations - it might not be uniform.
In many places $\subset$ implies a proper subset, while $\subseteq$ implies a possibly improper subset. In books you will find the definition somewhere, but in questions here... you just have to "guess" the right definition from the context.
Personally I am always in favor of clarity (when possible), $\subseteq$ and $\subsetneq$ are my choice of symbols. One of my teachers even takes $\subseteqq$ and $\subsetneqq$ for even greater clarity.
A: That depends on the author, see here.
A: Different people use different conventions.  Some people use $\subset$ for proper subsets and $\subseteq$ for possible equality.  Some people use $\subset$ for any subset and $\subsetneq$ for proper subsets.  Some people use $\subset$ for everything, but explicitly say "strictly proper" in words when they feel it matters.  I do not believe that there is a consensus for the meaning of $\subset$.  My own personal advice is to use $\subseteq$ and $\subsetneq$ when you care to be precise, and $\subset$ when you are feeling lazy.  
A: My convention has always been $\subset$ is strict and that $\subseteq$ is nonstrict.  This maintains parallelism with $<$ and $\le$.  
I have seen $\subset\subset$ in the comments. I have seen it used in this context.
Write $K\subset\subset G$, when $K$ is compact and $G$ is open.
